Post-buckling behaviour of a growing elastic rod

We consider mechanically-induced pattern formation within the framework of a growing, planar, elastic rod attached to an elastic foundation. Through a combination of weakly nonlinear analysis and numerical methods, we identify how the shape and type of buckling (super- or subcritical) depend on material parameters, and a complex phase-space of transition from super- to subcritical is uncovered. We then examine the effect of heterogeneity on buckling and post-buckling behaviour, in the context of a heterogeneous substrate adhesion, elastic stiffness, or growth. We show how the same functional form of heterogeneity in different properties is manifest in a vastly differing post-buckled shape. Finally, a fourth form of heterogeneity, an imperfect foundation, is incorporated and shown to have a more dramatic impact on the buckling instability, a difference that can be qualitatively understood via the weakly nonlinear analysis.